The objective of this project is the further development of new non and semi-parametric methods for the analysis of health studies with missing data and censored data. The main emphasis is in the development of inferential methods that make the most effective use of the observed data under models that make minimal assumptions about the full data law and the censoring or missingness mechanism. The first aim is to develop a general methodology for the analysis of non-ignorable informative right censored data with possibly competing types of censoring that will allow the analyst to appropriately adjust for informative censoring due to measured factors while simultaneously quantifying the sensitivity of the inference to residual dependence of the outcome of interest and censoring due to unmeasured factors. The methods will be applied to the analysis of failure time data, quality of life adjusted survival data, medical cost data and, more generally, to inference about a function of an increasing stochastic process with a randomly stopped censored stopping time. The second aim is to develop methods of inference about the mean of a K-sample U-statistic with missing data. These methods will be applied to derive corrections to the Mann-Whitney distribution free test and to estimate the area under the receiving operator characteristic curves of diagnostic tests obtained from studies with selective ascertainment of disease status. The third aim is to develop new pattern mixture models for longitudinal incomplete data which assume semi or non-parametric models for the distribution of the missing data in each stratum of non-response. The fourth aim is to develop semiparametric methods for the analysis of complex study designs with follow-up of non-respondents. The fifth aim is to develop the theory of doubly-robust estimation and to construct doubly-robust estimators in missing data problems. Doubly robust estimators are attractive because they remain consistent and asymptotically normal under misspecification of either (but not both) the model for the missingness mechanism or the model for the distribution of the full data.